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An overview manual for the GRAVSOFTï¼
Kort
&
matrikel
styrelsen. Heiskanenï¼W. A. and moritzï¼H.ï¼1967. Physical Geodesyï¼san Fransiscoï¼W.H. Freeman ...
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, , ,
China Ocean Engineering Vol. 23 No. 4 pp. 679 - 694
?
2009 Chinese Ocean Engineering Society ISSN 08905487
Development of Precise Geoid Model for the Establishment of Consistent Height System in Geoga Grand Bridge Construction Area
,
DongHa LEE HongSic YUN1 and He HUANG
,Architectural and Environmental System Engineering,Sungkyunkwan University, Suwon 440 746,Korea (Received 1 September 2008;received revised form 24 April 2009;accepted 13 June 2009)
Department of Civil
ABSTRACT The national benchmarks on islands were mostly established by trigonometric leveling in Korea. This method results in inaccuracy which is a serious problem in Geoga Grand Bridge construction work that tried to link the mainland and the islands. The Geoga Grand Bridge PusanGeoje fixed link project was selected as the study area a huge construction work in Korea that will connect the mainland Pusan and an island Geoje island . However the orthometric heights issued at benchmarks JINH and GOEJ were not consistent because they did not refer to the same zero point which would make the linking of the sections problematic. This paper introduces the precise local geoid as a vertical datum for the construction area in order to establish a consistent height system. To determine the precise local geoid for the con struction area we firstly developed a precise gravimetric geoid for Korea and its adjoining seas as a whole. This gravimet ric geoid was developed by use of all available gravity data including surface and satellite data on land and on the ocean. The gravimetric geoid was computed by spherical fast fourier transform with modified Stokes′ kernels. The removerestore technique was used to eliminate the terrain effects by use of the RTM reduction and to determine the residual geoid by combining the GGM02S / EGM96 geopotential model freeair gravity anomalies and highresolution DEM data. Finally the gravimetric model was fitted to the geoid heights obtained from GPS and tide observations N GPS / Tide by least square collocation to provide the final GPSconsistent local precise geoid model. The postfit error std. dev. of the final geoid to the N GPS / Tide derived from GPS and tide observations was ± 2 . 2 cm for the construction area. We solved the height inconsistency problem by calculating the orthometric height of the benchmarks and the control points using the final geoid model. Also the highly accurate orthometric height was estimated through the GPS / leveling technique by applying the developed local precise geoid. Therefore the precise local geoid is expected to improve the quality of the construction procedure of the Geoga Grand Bridge.
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Key words precise local geoid consistent height system gravity observations GPS / Tide observations least squares fit ting
1 . Introduction
,
In Korea national benchmarks were established and are maintained by the National Geographic
( ) poses:surveying,mapping,research and construction .
Information Institute NGII . NGII issues the exact orthometric heights of benchmarks for various pur
,the benchmarks are important points since they are used as a reference of orthometric height in construction work . Therefore,the al titude of these benchmarks should be measured exactly by spirit leveling . However,the benchmarks lo Especially
cated on islands were mostly established by trigonometric leveling method . This method yields inaccu
:
1 Corresponding author. Email yhs@ geo. skku. ac. kr
680
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DongHa LEE et al . / China Ocean Engineering 23 4
, cal angle (the incident angle of prism)of 10° (Rho et al . ,2001). This quantity can be a serious rate leveling results up to 5 . 6 mm / km when using EDM and observing simultaneously with the verti problem in the bridge construction work .
( )
( )
Fig. 1 . Relationship between the height systems on the mainland JINH and on the island GEOJ .
,
(
)
The ongoing construction project Geoga Grand Bridge PusanGeoje fixed link project to con
( ) ( ) , tions:4 . 5 km cablestayed girder bridge,3 . 7 km immersed tunnel,and 25 . 7 km connection road . For linking of the sections,control points were installed at each section and their heights were comput ed by applying the orthometric height of the national benchmarks,JINH and GEOJ . However,some difference over 30 cm in the height system as shown in Fig . 1 between the mainland (JINH)and the island ( GEOJ)was revealed in the process of the route surveying for road design by Daewoo E&C Co . Ltd . ( Daewoo E&C Co . Ltd . ,2003). This difference regarded as the height error between the bench marks (JINH and GEOJ)were caused by inaccurate leveling result of benchmark (GEOJ)on the is land,which means that the height system between the mainland and the island were not consistent . In other words,the orthometric heights issued at benchmarks (JINH and GEOJ)were not referred to the same zero point,which would make the linking of the sections problematic . Different approaches have been studied for fixing the height inconsistency problem among different height systems in the context of height system unification (Zhang et al . ,2009). The main issue of height system unification is to determine the potential difference among different height systems ( Rum mel,2000). Geodetic leveling and ocean leveling method are wellknown for the height system unifica tion . The geodetic leveling is a very highly accurate method to determine the potential difference,but it cannot be used to unify the different height systems separated by oceans . Another method is ocean leveling (Ekman,1999)that can be used to determine the potential difference between different height systems across ocean,but it has a low accuracy with respect to the geodetic leveling method (Zhang et al . ,2009). A new method for unifying different height systems as proposed by Bura et al . ( 1999) is to use the GPS / leveling data for a local area in order to determine the potential difference between local precise geoid and mean sea surface,which is defined by a tidal gauge observations (Bura et nect the mainland Pusan and the islands Geoje island in Korea consists of three different sec
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, )
681
,
,
al . 2004 Zhang et al . 2009 . In principle if we can find the local mean sea surface which is
, mean sea surface for determining the potential differences among different regions (Fei and Sideris, 2001;Li et al . ,2003;Zhang et al . ,2009;Amos and Featherstone,2009). Therefore,Daewoo E&C Co . Ltd . recognized that the height differences of both benchmarks (JINH and GEOJ)were caused by the inconsistent height system problem and it is necessary to deter irrelevant to the reference height system we can make the comparison of local precise geoid with the
mine the local precise geoid model for the consistent height system of the construction area of the Geoga Grand Bridge . Fig . 2 shows the distribution of benchmarks and control points in the construction area of the Geoga Grand Bridge .
()
Fig. 2 . Location of the construction area left and distribution of benchmarks
( )
and control points in the construction area right .
(
)
This paper describes that a new vertical datum the consistent height system for the construction area was established by use of precise local geoid model to minimize the cost and time consumption problem due to the height errors from the trigonometric leveling method . To determine a precise local
,
geoid for the construction area we firstly developed a precise gravimetric geoid for Korea and its ad
,
joining seas as a whole . This gravimetric geoid was developed by use of all the available gravity data
including surface and satellite data on land and ocean area . The existing surface gravimetric data are
, ,
not sufficiently dense for the determination of a precise local geoid thus we perform a relative gravi metric survey at sixtyone points with 2 km spaces approximately in the construction area . The gravimetric geoid was computed by spherical fast fourier transform
(
, )
(FFT)with modified
Stokes′ kernels Forsberg et al . 1993 . The removerestore technique was used to eliminate the ter
( ) ( , ) the residual geoid by combining a geopotential model,freeair gravity anomalies,and highquality res
rain effects on land by the residual terrain model RTM reduction Forsberg 1985 and to determine olution DEM data .
,
For the construction area the accuracy of the gravimetric geoid computed from the difference be
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682
( ) ( ) was determined . The geoid difference (N - N )had a mean value of 0 . 586 m and a standard deviation of 4 . 8 cm;these results were quite satisfactory to give the large expected changes in the gravity field,the lack of accurate gravity data offshore,and the noise in the tidal height transfer . As such,this gravimetric model was subsequently fitted to geoidal heights from GPS / leveling and tide observations by use of least square collocation,to provide the final GPSconsistent local precise Kuroishi et al . ,2002). The postfit error (std . dev . )of the final geoid to the N geoid model ( derived from GPS and tide observations was ± 2 . 2 cm,and the final geoid could be used to obtain consistent heights for the construction area . Therefore,the inconsistent height system problem (as a linking problem)was solved by calculating the orthometric height of the benchmarks and the control
tween gravimetric geoidal height N grav and geoidal height from GPS and tide observations N GPS / Tide GPS / Tide
grav
GPS / Tide
points using the final geoid model and ellipsoidal height from GPS observations .
2 . Gravimetric Geoid of Korea 2.1
Computational Formulae for Gravimetric Geoid Determination
,
In the removerestore technique the gravimetric geoidal N height is split into three parts N = N GM + N RTM + N res
,
(1)
where N GM is the lowfrequency part of the geoid obtained from a geopotential model . For computation
, ( ,2005) model and the EGM96 (Lemoine et al . ,1996)model was used . EGM96 was determined from a com bination of satellite tracking data,satellite altimetry in the oceans and mean gravity anomalies on land . The new satellite GRACE GGM02S field was used up to 90 degrees to improve the EGM96 model,so that GGM02S could be used exclusively to 90 degrees,and EGM96 above 90 degrees,with the linear merging of spherical harmonic coefficients . Grids of gravity ( g )and geoid (N )from the Δ GGM02S / EGM96 model were computed for the area 32° ~ 40°N,123° ~ 132°E. N ,the highfrequency part of the geoid,is the terrain effect on the geoid generated by the residual terrain model ( RTM)method (Forsberg et al . ,1981). The RTM terrain reduction evaluates the gravitational effects of the mass anomalies relative to a mean elevation surface (Forsberg,1985). The mean elevation surface is determined by movingaveraging filtering by local DEM data (Tcherning et al . ,1994). The mean elevation surface corresponds in principle to the topographic signal already present in the spherical harmonic reference model . In the case of Korea,the mean DEM surface of ap prox . 75 km ( 42′ × 58′)resolution was used . For the calculation of the terrain effect by RTM,we used local DEM on a 100 m resolution grid, derived from original DEM on a 30 m grid (see Fig . 3). The original DEM was produced from digital topographic maps drawn on scale 1: 25000 and provided directly by Environmental Geographic Informa ,Korea (see,http:/ / egis . me . go . kr / ). The height value in local DEM distribut tion System (EGIS) of the reference geoid undulation N GM the combined model of the GGM02S Tapley et al .
GM
GM
RTM
ed from 0 to 1950 m and the standard deviation was about ± 12 . 0 m by comparing with the orthometric heights of GPS / leveling data .
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(
, ):
The expression for the geoid effect of the topography is given as Omang et al . 2000
( ,) z = h (x ,y ) ( 槡 xQ
ρ ∫∫∫ γ
) (2) where G is the gravitational constant and ρ is the density constant of topographic mass . (x ,y ,z ) is the coordinate of DEM grid point (Q )and (x ,x ,x )is the coordinate of gravity station (P ). In practice,N can be computed in the frequency domain by Fourier methods,using a linear term (corresponding to the residual topography being condensed on the geoid)(Forsberg,1985) Gp 1 (3) N = γ(h - h ) s , where s is the horizontal distance between the gravity station and the DEM grid point,and denotes the 2D convolution,for details in Forsberg ( 1985). The corresponding RTM effects on gravity data (Δg )were computed directly by space domain prism integration using the dense height data . N RTM =
G
∞
∞
-∞
-∞
dxQ dyQ d zQ
z=h x y ref
- xP
) + (y 2
Q
- yP
) + (z 2
Q
- zP
2
Q
P
P
Q
Q
P
RTM
RTM
ref
RTM
Fig. 3 . Local DEM data with 100 m resolu
( :)
tion unit m .
,
,
N res the mediumfrequency part of the geoid is the residual geoid computed from
,
, R , g S( = Δ Ψ)dσ 4π γ σ
g Δ
res
by use
of Stokes integration extending in principle the integration to cover the Earth . N res where
g Δ
res
(4)
res
is the residual freeair anomaly that remains in the gravity data after the contributions of
the residual terrain effect
g Δ
RTM
and the global field
g Δ
GM
are subtracted . The indirect effect on gravi
( ) its effect is only a few centimetres on the gravimetric geoid computation (Corchete et al . ,2005). Thus,in this study,the secondary indirect effect has been neglected . The function S ( Ψ)is Stokes function,Ψ being the spherical distance . ). (5) S( log(sin Ψ+ sin Ψ + 1 - 5cosΨ - 3cosΨ Ψ)= sin(1Ψ)- 6sin Ψ 2 2 2 2 To prevent the influence of local gravity on the longest wavelengths (Vaníˇcek and Featherstone, ) 1998 ,the modified Stokes kernel S ( Ψ)is used . The technique is the modified WongGore ty or secondary indirect effect is not considered in most of the cases because it has small values and
2
mod
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(
, ),in which the modified kernel S (Ψ)has the following expres
method Wong and Gore 1969
mod
sion . N2
(Ψ)= S(Ψ)- ∑α (n)2nn -+ 11 P cos(Ψ), (n)increase linearly from 0 to 1 between degrees N and N where the coefficients α S mod
(6)
n
n=2
1
1
2
for 2 n N 1
N 2 - n = N2 - N1 0
(n) α
,… ,N
for
N1 n N2 . n = 2
for
N2 n
(7) ,
The coefficients N 1 and N 2 are determined by trial and error or by experience and represent a tradeoff between full use of the satellite fields for n > N 1 and full use of the local gravity data n >
(
)
)
(
N 2 . The task of optimization is typically to find a set of N 1 and N 2 for which the fit between the gravimetric geoid and the geoidal heights derived from GPS / leveling data is the best in the least squares
(
, )
,
sense Iliffe et al . 2003 . By these optimization the values of N 1 = 80 and N 1 = 90 are determined for the final computation of residual geoid .
(
The residual gravity is transformed into the residual geoid by multiband spherical FFT Forsberg
, ),which provides a virtually exact implementation of Stokes formula on a sphere . The geoid is obtained by a number of bandwise operations of the form ,Δ )[Δg (φ,λ )sinφ]= F [F(S)F(Δg sinφ)] (8) N = S( Δ λ φ et al . 1993
res
-1
res
res
where F is the twodimensional fourier transform operator . F
(Δg)=Δg(x ,y)e (
- i kxx + kyy
)dxdy ,
(9)
where kx and ky are the wave numbers and 100% zero padding is used in this study . The general de
( ) The outcome of the removerestore technique is a gravimetric geoid,referring to a global height datum approximately,which fits the Korean vertical datum,and minimizes the possible systematic er rors in long wavelength (lowfrequency)part of gravimetric geoid due to errors of geopotential model and / or truncation procedures ( Kuroishi et al . ,2002). This geoid must be fitted to the GPS control in ,de the final geoid determination step . The software package GRAVSOFT (Tscherning et al . ,1994) veloped at the Danish National Space Center and University of Copenhagen,is used for all computa tions in this study (Forsberg et al . ,2003). tails of the FFT methods were given by Schwarz et al . 1990 .
2.2
Gravity Data in Korea
Several gravity data sources for onshore Korea from various national and international agencies
,
were evaluated in preparation for the geoid determination . From NGII 2857 points gravity data were
( )
obtained and 2000 points gravity data from Bureau Gravimetrique International BGI were used . The
,
,
ocean gravity data derived from ERS GEOSAT and TOPEX satellite altimetry of the ocean areas from
(
, ),were used . There were uncertainties in the datum and systems of the reference for some of these gravity data,and errors were confirmed by bias compar
KMS2002 global solution Andersen et al . 2005
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isons with the GGM02S / EGM96 reference model .
,
, was carried out . This survey used a Lacoste and Romberg G200 gravimeter,tying the relative gravity measurements into an absolute station DAEJ with the reference gravity value,g :979832 . 444 mgal . The gravimetric survey took place during the period 12 ~ 16 Sep . 2005,observing a total of 70 observations at 61 points . A leastsquare adjustment was carried out on the observations,giving a stan dard error of 0 . 073 mgal for a single observation . A single instrument tare of 0 . 24 mgal was detected on 13 Sep . between two points,so the points in this sequence might have an error of this magnitude . However,this magnitude of error has negligible influence on the geoid results,and all observations are However the existing surface gravimetric data were not sufficiently dense so a gravimetric survey
used for gravimetric geoid computation .
Fig. 4 . Location of new gravity points around construction area. All points were surveyed by precise DGPS relative to the fundamental
( )
benchmark at Pusan JINH .
(λ , )and orthometric heights (H )of gravity points approximately,we performed a differential GPS ( DGPS)survey relative to the fundamental benchmark at Pusan (JINH)with gravimetric survey simultaneously . GPS observation period for data sets are be tween 1 ~ 2 hours . Trimble Geomatics Office (TGO)software was used for baseline processing of the whole GPS data and network adjustment . The network adjustment is based on a local datum as the fixed height of JINH and geographical coordinates of two GPS permanent stations (JINJ,PUSN)locat ed near the study area . According to the adjustment results,the horizontal (or geographical)and verti cal (or orthometric)coordinates of the gravity points are determined with the precision smaller than 1 cm and 5 cm,respectively . The gravity data was evidently affected by some gross errors . Only a few large outlier values were removed with 95% significance level,corresponding to the areas mainly close to the construction area To determine the geographical coordinates
and in the BGI and KMS data . The gravity data were reduced for the reference field and RTM terrain effects . Terrain effects were computed from the 100 m resolution height data using prism integration . The statistics of the gravity re ductions are shown in Table 1 . It is uncertain whether the biases in the NGII data are indicative of residual systematic errors .
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Fig. 5 . Location of gravity points used for gravimetric geoid determination. Background is the reduced gravity data with GGM02S / EGM96 and RTM.
Table 1
Statistics of data reductions for gravity data after removal of gross error Data set
No. of pts.
:
unit mgal
Mean
Std. dev.
Min.
Max.
21 . 86
23 . 17
- 26 . 05
176 . 10
- 9 . 37
21 . 12
- 65 . 11
141 . 93
Reduced for GGM02S / EGM96 and RTM
- 3 . 51
11 . 74
- 52 . 23
98 . 20
BGI data
24 . 98
16 . 96
- 16 . 60
147 . 20
- 9 . 75
16 . 92
- 44 . 99
122 . 98
Reduced for GGM02S / EGM96 and RTM
0 . 04
12 . 21
- 39 . 55
123 . 67
New gravity data
22 . 95
4 . 67
4 . 83
31 . 98
- 0 . 94
4 . 64
- 11 . 22
9 . 23
3 . 46
4 . 12
- 9 . 03
9 . 78
NGII data Reduced for GGM02S / EGM96
2759
Reduced for GGM02S / EGM96
2000
Reduced for GGM02S / EGM96
61
Reduced for GGM02S / EGM96 and RTM
2.3
Practical Computation of Gravimetric Geoid
The gravimetric geoid was computed from the reduced gravity data by gridding with least squares
,
( )
, tion of the terrain effects and GGM02S / EGM96 model effects . Data were gridded in the area 33 ~ 39° N,124 ~ 131°E,at a basic grid spacing of 0 . 0125° × 0 . 016667° ( 0 . 75′ × 1′)in latitude and longi tude . The comparison of our results with the results of Yun ( 1995)indicates that 2D multiband spher collocation twodimensional 2D multiband spherical FFT transformation followed by the restora
ical FFT with 4bands has given the best solution for Korea . The FFT was performed by use of 100% zeropadding to limit the periodicity effects . The 100% zeropadding puts zeros around the values of
(
),practically doubling the dimensions (Yun,1999).
the original field input matrix
The final gravimetric geoid in Fig . 6 was computed by 2D spherical FFT with 4bands and 80 ~
,
90 modified WongGore kernel . Figs . 7 and 8 show the geoid effects of the local gravity data and the effect from the topography .
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687
Fig. 6 . Gravimetric geoid of Korea in global vertical reference system
)
(unit:
m . Location of GPS leveling data
(
shown with cross all GPS / Leveling
)and triangle (GPS / Leveling )
data
data around construction area .
Fig. 7 . Geoid effects from the transformation of the
Fig. 8 . Restored geoid effects of the residual
combined residual gravity data (unit:m).
( :)
terrain model unit m .
Table 2 shows the statistics of comparison of gravimetric geoid and geoid obtained from two GPS / levelling data sets in Fig . 6 . The Korean geoid model has a reasonable but not the best fit to GPS . This is probably due to the systematic errors and outliers in both gravity and GPS / leveling data . Also
,
shown is the fit in a more localized area around the construction area used in the final fit of the geoid
(
)
to GPS and tide GPS / Tidegauge observations in the following section . Table 2
:
Statistics of comparison of gravimetric geoid and geoid from GPS / leveling unit m GPS data set
GPS / Leveling data (all) GPS / Leveling data (around construction area)
No. of pts
Mean
Std. dev.
Min.
Max.
447
0 . 85
0 . 16
- 0 . 21
2 . 00
74
0 . 87
0 . 12
0 . 53
1 . 09
3 . GPS and Tide Observations The tide observations were carried out primarily to achieve an independent control of the gravimet
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688
,
ric geoid heights across the strait . The measurements were done over a few days using repeated local measurements to the instantaneous sea surface in connection with the layout of the tide gauges . Three
( ,
,
)
( )
temporary tide gauges JIN1 GEO1 GEO2 and one permanent tide gauge GADO were used . Fig . 9 shows the location of the tide observation points .
Fig. 9 . Location of tide observation points.
GADO is a permanent tide gauge for the hydrographic service . This tide gauge provides an unin terrupted time series for the duration of all other tide gauges . A 24hr error in the time tagging was ob
,
vious and corrected prior to the use of data . JIN1 GEO1 and GEO2 were observed by a temporary tide
(
),as the same period 280 . 0 ~ 286 . 0 in Julian day (JD). JIN1 provided an uninterrupted time series in the period,281 . 091 ~ 285 . 174 in JD. These nois es in the time series of JIN1 were due to waves and boats . GEO1 apparently moved at two periods,JD: 282 . 22 (# 1)and 283 . 62 (# 2). The amount of movement was not well defined,the data were only used in the period,JD:280 . 245 ~ 282 . 200 . GEO2 was also moved at two periods,JD:282 . 24 (# 1)and 284 . 95 (# 2). These movements were likely attributed to curious people taking the sensor out of the water and throwing it back again . The first break was clearly fixable (offset of 19 . 8 cm). The useful data for this tide gauge were therefore JD:281 . 238 ~ 284 . 952 . gauge during 6 days 7 Oct . 2005 until 12 Oct . 2005
The height of the GPS reference point above the mean sea level for a time interval
can be expressed as:
t= t Δ
,
- t1
(10)
HΔ t = H m + t m - tΔ t
; ,
2
where HΔ t is the mean sea level during the period H m the measured height difference from the GPS
;,
;
,
reference point t m the average tidal readings corresponding to the measurements H m and tΔ t the average tide gauge readings during the longer period
;
,
t . The height HΔwill be affected by longperi Δ t
,
od tides however the height differences between stations correspond to orthometric height differences since the longperiod tides may assumedly be common to all stations .
,
,
Based on the useful tidal intervals with the GADO tide gauge used as a reference and the pairs of the heights above sea level for the relevant intervals
t were determined,as shown in Table 3 . The Δ
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Fig. 10 . Sea level height variation from tidal observations.
()
Dterm in Table 3 indicating the term D = H m + t m in Eq . 10 should be constant . Table 3
Heights above local sea level for specific period intervals No. of
Mean of
Std. dev. of
H m meas
Dterm
Dterm
GADO
16
3 . 34
0 . 014
GADO
GADO
JIN1
Station
()
Interval JD
No. of
HΔ t
tidal meas
(m)
Comments
281 . 091 ~ 285 . 174
5860
2 . 316
Use with JIN1
280 . 245 ~ 282 . 200
2806
2 . 285
Use with GEO1
281 . 238 ~ 284 . 952
5330
2 . 323
Use with GEO2
6
3 . 175
0 . 022
281 . 091 ~ 285 . 174
2940
1 . 933
GEO1
6
4 . 480
0 . 049
280 . 245 ~ 282 . 200
281
3 . 308
GEO2
6
4 . 552
0 . 023
281 . 238 ~ 284 . 952
5343
1 . 781
The orthometric heights in the Korean height system can subsequently be found from the height
,
differences of Table 3 . For this purpose the heights were transferred from the fundamental benchmark JINH to JIN1 by use of the gravimetric geoid . This transfer was assumed to be accurate since the base
,
line length was very small . The orthometric height of JIN1 was subsequently transferred to GADO GEO1 and GEO2 by use of the relevant height differences in Table 4 .
,
The GPS coordinate of JINH defines the datum of the GPS system. Similarly the N GPS / Tide de
,
,
,
fines the local geoid datum i . e . the geoid value when subtracted from GPS will give orthometric heights in the Korean national height system. The comparison of Tables 3 and 4 shows that the local sea level in the construction area is close to the zero height of Korea . GADO has thus an orthometric
; , 2 . 323 m. Therefore,the difference in height was 25 cm,smaller than the typical seasonal variations of height of 2 . 571 m however it has a height above the mean sea level for four days in October 2005 of
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sea level . The last column in Table 4 gives an estimate of the geoid difference between the gravimetric geoid
(
)
N grav and the independent geoid information N GPS / Tide derived by
(
N GPS / Tide = h GPS / Tide - H levelling .
)
(11)
,
The geoid difference N GPS / Tide - N grav has a mean value of 0 . 586 m and a standard deviation of 4 . 8 cm. The standard deviation is quite satisfactory to give the large expected changes in the gravity
,
,
field the lack of accurate gravity data offshore and the noise in the tidal height transfer . The mean
(
value of the geoid difference expresses the difference between the gravimetric geoid referring to a glob
)
()
al datum and the GPS geoid from Eq . 11 which by definition refers to a local sealevel datum. Table 4 shows that the benchmark GEOJ on the Geoje island does not fit the geoid and tidal height
;
,
(
)
transfer therefore there is likely an error in one of the heights either leveled height or GPS height
,
for this station . Therefore the N GPS / Tide value of GEOJ is not used in the fitting for final geoid deter mination . Table 4
, ellipsoidal heights from GPS survey and derived geoid heights (N
Heights in the Korean height system of the tide gauge points with corresponding
)
GPS / Tide
N GPS / Tide
N GPS / Tide
= h- H
- N grav
28 . 414
28 . 952
0 . 538
2 . 188
28 . 394
28 . 932
0 . 538
31 . 562
2 . 571
28 . 392
28 . 991
0 . 599
128 . 69680
32 . 569
3 . 594
28 . 327
28 . 975
0 . 648
34 . 89061
128 . 69918
30 . 982
2 . 029
28 . 393
28 . 953
0 . 560
34 . 88940
128 . 69049
48 . 090
18 . 763
28 . 399
29 . 327
0 . 928
Ellipsoid
Orthom.
height h
height H
128 . 81728
36 . 489
7 . 537
35 . 09310
128 . 78932
31 . 120
GADO
35 . 02416
128 . 81087
GEO1
34 . 98064
GEO2
GEOJ
Point
Lat.
Lon.
JINH
35 . 10001
JIN1
,
,
N grav
Comments
Datum point Ortho h from JINH by geoid Ortho h from JIN1 by tidal Ortho h from GADO Ortho h from GADO Geoje island Bench Mark
4 . Final Geoid Model for the Consistent Height System
,
The final geoid of the area was determined by fitting the N GPS / Tide values of Table 4 together with
,
a subset of the GPS / leveling data points as shown in Fig . 7 . By use of the geoid information from
,
GPS / leveling or the GPS / Tide observation links together with geoid from gravimetry the long wave
,
length geoid errors can be suppressed and the inherent datum differences eliminated . It is essential
,
when computing GPS geoid heights that both leveling and GPS heights are as error free as possible
, (),2009,679 - 694
DongHa LEE et al . / China Ocean Engineering 23 4
691
otherwise these errors will creep into the fitted geoid . The fitting of a gravimetric geoid to a set of GPS geoid heights entails modelling the difference signal
ε= N
, ε
GPS / Tide
(12)
- N grav
and adding the modelled correction signal to the gravimetric geoid . In this way a new geoid grid tuned to the leveling and GPS datum can be obtained . The method of least squares collocation is used for es
,
timating the trend and modelling the residuals . For trend estimation we used the 4parameter model
(Heiskanen and Moritz,1967)of the form, (13) cosλ sinλ ′, ε= cosφ α+ cosφ α+ sinφ α+ α+ ε where,parametersαtoαcorrespond to the geoid effects of a Helmert transformation . ε ′ is the resid ual geoid errors modeled by least square collocation of the form, ^ = C C { (14) ′ }. ε ε In the collocation process,a covariance function must be assumed for the residual geoid errorsε ′ as a function of distance s . We used the secondorder Markov covariance function of the form (Iliffe et , al . ,2003) (15) C (s )= C ( 1 +α s )e α. Such a covariance function is characterized by the variance C and correlation length s ,which, in turn,determine the degree of fit and the smoothness of the interpolated geoid error . The constant α is the only quantity to be specified,with C automatically adapted to the data . In the selection of the correlation length and noise of observed errors,the user has a wide range of selection options . Either a strong fit to the GPS data,or a more relaxed fit,can diminish the impact of possible errors in the GPS data (Forsberg et al . ,2003). We determined a correlation length of 40 km by a standard LSC proce dure ( Moritz,1980)and 1 cm a priori GPS noise was assumed . The geoid correction signal is shown 1
1
2
3
4
4
-1 xx sx
- s
0
0
1/2
0
in Fig . 11 .
(
)
The postfit error std . dev . of the final geoid to the N GPS / Tide derived from GPS and tide obser vation is ± 2 . 2 cm for the construction area and ± 4 . 3 cm for the other areas around the construction area . Table 5 shows the postfit statistics for the local datum. The final local precise geoid is shown in Fig . 12 . Table 5
:
Postfit statistics for the local datum Data set
N Tide derived from GPS / tide gauge fix
(
)
points in construction area
N GPS derived from GPS / Leveling data
(around construction area)
unit m
No. of pts
Mean
Std. dev.
Min.
Max.
5
0 . 001
± 0 . 022
- 0 . 023
0 . 024
74
- 0 . 002
± 0 . 043
- 0 . 112
0 . 118
5 . Results and Conclusions
,
In the present study we developed a local precise geoid model to solve the problem of the incon
, (),2009,679 - 694
DongHa LEE et al . / China Ocean Engineering 23 4
692
Fig. 11 . Geoid correction signal from local GPS / level
( :)
Fig. 12 . Final geoid for the construction area of Geoga
ing / tide gauge data unit m .
,
Grand Bridge from gravity GPS and tide gauge
,
measurements with contour interval of 2 cm.
sistent height system occurring in the construction area of the Geoga Grand Bridge caused by the large
,
difference over 30 cm in the orthometric height of the benchmarks JINH and GEOJ . For the determi
, , for Korea and its adjoining seas as a whole . This gravimetric geoid was developed by using all avail able gravity data,including surface and satellite data on land and ocean area . The gravimetric geoid was computed from the reduced gravity data by gridding with least squares collocation,2D multiband spherical FFT transformation,and restore of the terrain effects and GGM02S / EGM96 model effects . Data were gridded in the area 33 ~ 39° N,124 ~ 131° E,at a basic grid spacing of 0 . 0125° × 0 . 016667° ( 0 . 75′ × 1′)in latitude and longitude . The final gravimetric geoid was computed by 2D spherical FFT with 4bands and WongGore modification to degree band 80 ~ 90 . Finally,this gravimetric model was subsequently fitted to the GPS / leveling and tide observations by least square collocation,to provide the final GPSconsistent local precise geoid model . For the col location process,we determined a correlation length of 40 km for a secondorder Markov covariance function and 1 cm a priori GPS noise was used . The postfit error (std . dev . )of the final geoid to the nation of a precise local geoid of the construction area we first developed a precise gravimetric geoid
N GPS / Tide derived from GPS and tide observations was ± 2 . 2 cm in the construction area of the Geoga
,
Grand Bridge . Based on the final geoid the new orthometric height was computed at the benchmarks and control points in the construction area . Table 6 shows the results of the new orthometric heights and the differences . The differences between the old orthometric height for the bridge design and the new orthometric
,
,
height from the final geoid are 0 . 007 m at JINH which is the mainland benchmark and 0 . 342 m at
,
GEOJ which is the island benchmark . The modification of orthometric heights for bridge design is necessary for accurate and efficient construction work .
, (),2009,679 - 694
DongHa LEE et al . / China Ocean Engineering 23 4
Table 6
New orthometric heights of benchmarks and control points from the final geoid
Orthoheight Points B. M.
Control Point
B. M.
693
JINH
Lat.
Lon.
Orthoheight for
(Deg. )
(Deg. )
bridge design A
35 . 10001
128 . 81728
8 . 6675 m
New orthoheight
Difference
8 . 671 m
0 . 007 m
( ) from final geoid (B) (A - B)
CP101
35 . 02882
128 . 81902
24 . 121 m
24 . 438 m
0 . 317 m
CP116
35 . 02146
128 . 80612
15 . 873 m
16 . 218 m
0 . 345 m
CP110
35 . 01604
128 . 76247
9 . 628 m
9 . 989 m
0 . 361 m
CP104
35 . 01506
128 . 71758
80 . 318 m
80 . 665 m
0 . 347 m
CP100
35 . 02117
128 . 71770
12 . 649 m
12 . 975 m
0 . 326 m
GEOJ
34 . 88940
128 . 69049
18 . 7630 m
19 . 105 m
0 . 342 m
,
In the present study we solve the inconsistent height system problem by calculating the exact or thometric heights of the benchmarks and the control points by using final geoid model . The final geoid
,
can be used to give consistent heights in the construction area . Also the precise orthometric height is
,
estimated through the GPS / leveling technique by applying the final geoid model . Therefore the pre
,
, procedure of the Geoga Grand Bridge . However,the final geoid model cannot be used directly for an accurate bridge construction,such as the precision loft of bridge pier and pier cap etc . ,because it has a low accuracy (± 2 . 2 cm)with respect to the allowable accuracy of accurate bridge construction . In order to use a local geoid model directly for the accurate bridge construction,it is necessary to enhance
cise local geoid which is developed in this study is expected to success the quality of the construction
the accuracy of geoid model up to the submm level by longterm tidal observations and additional grav ity observations in the construction area of the bridge . References
,
, , : , , (): , , , , , ( ) ,: , , , , , , ,, , , , , , , , , , , , , (): , , ,, ,, , , , , , , , (): , , , , , : , , (): , , , ( ) , , , , (): , , , ,: , , , , , ,
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